Show that B = {x^2, (x -1)^2, (x + 1)^2}is a basis for P2
I'm not even sure how to do this, so any help would be great!
Well, you want to show that any polynomial in $\displaystyle P_{2}$ can be written as a linear combination of elements of $\displaystyle B.$ That is, for an arbitrary element $\displaystyle ax^{2}+bx+c\in P_{2},$ you want to show that
$\displaystyle ax^{2}+bx+c=tx^{2}+u(x-1)^{2}+v(x+1)^{2},$
where $\displaystyle t,u,v$ are constants to be determined in terms of $\displaystyle a,b,c.$ Does that give you some direction?
Well, the equation exhibited in post # 2 has to hold for all values of x. If you plug in those three values, one at a time, you get three fairly simple equations for the three unknowns. Then you can solve for them. And, assuming you get even one solution, you will have proven what you need to prove.
Substitution, or elimination is probably the best way to go.